Who Was the Greatest of All-Time? a Historical Analysis by a Complex Network of Professional Boxing

Home , Andre Ward, Archie Moore, Barney Ross, Benny Leonard, Bernard Hopkins, Boxing

Journal of Complex Networks (2020) 1, Advance Access Publication on 29 February 2020 doi: 10.1093/comnet/cnaa009

Who was the greatest of all-time? A historical analysis by a complex network of professional boxing Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 Adam G. Tennant† Department of Engineering, 2030 Business and Engineering Center, University of Southern Indiana, 8600 University Boulevard, Evansville, IN 47712, USA †Corresponding author. Email: [email protected] Chase M. L. Smith Kinesiology and Sport Department, Health Professions Center 3092, University of Southern Indiana, 8600 University Boulevard, Evansville, IN 47712, USA and Jotam E. Chen C Department of Engineering, 2030 Business and Engineering Center, University of Southern Indiana, 8600 University Boulevard, Evansville, IN 47712, USA

Edited by: Ernesto Estrada

[Received on 16 October 2019; editorial decision on 23 January 2020; accepted on 3 February 2020]

This study seeks to examine and compare boxers throughout history creating a pound-for-pound list of the different ﬁghters. A PageRank algorithm was utilized to rank the boxers from the network to determine a list of the top 10 ﬁghters from 1897 to 2019. Two data sets were utilized, a truncated subset and a larger data set, to explore the impact of network size on the rank of boxers. Additionally, the researchers systematically varied the damping factor of the PageRank algorithm to determine the effects on the rankings. A discussion of the results includes a comparison of journalistic rankings and those from a points-based system from the respected boxing website BoxRec.

Keywords: boxing; PageRank; ranking.

1. Introduction Pugilistic historians for more than a century have engaged each other in a theoretical debate on who was the greatest of all time within the four corners of the boxing ring. Often these debates fall short due to a lack of evidence, these arguments must encompass wide time frames, multiple weight classes of boxers and a massive data set. The debaters must rely on speculative arguments such as boxer A’s speed was far superior to boxer B’s and this would have won the fight for them. These debates can be entertaining in a pop culture sense but only address the small data set of fighters that one person’s mind can bring to the debate. Additionally, they are tremendously biased to one individual’s preferences for style of fighting or even ethnic bias. The culture of sport in America is one that prides itself on undisputed, unquestioned, unanimous champions for both team sports and individual. Typically, fans can identify these champions from special events (e.g. Super Bowl, US Open, World Series, The Masters, etc.). While the boxing world contains

© The authors 2020. Published by Oxford University Press. All rights reserved. 2 A. G. TENNANT ET AL. title fights, fans are only privy to identifying an undisputed champion within a weight class. This leads many to wonder who is the best of the best. The pound-for-pound (P4P) argument is probably one of the most contentious in the sport and everyone seems to have an opinion on it [1]. This study attempts to breakout this debate, from speculation to more quantitative arguments by sharing preliminary findings of a sports data analytics approach. The researchers in this study ranked individual boxers through the PageRank algorithm and created a historical model of the complexity of the sport. The specifics of the Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 research are discussed in this study along with the results of the rankings. Although the science of boxing for the combatants is brutally simple, to hit and not get hit, the sport additionally offers a rich source of data for complexity science. Boxing, with its long history, has a depth to its data that can be explored with complexity science to yield insights that analysis from standard statistics has left in the dark. The researchers will explore whether Google’s PageRank algorithm sufficiently creates an all-time greatest P4P list regardless of weight class or historical time frame.

2. Background The depth and breadth of complexity science and network science provides insight into such diverse set of research topics as: cities [2–5], infrastructure [6], health [7], biology [8] and even friendship [9]. Sports is a topic in the field of data analytics that is just beginning to be explored in earnest. Cassady et al. [10] took the approach of using a quadratic-assignment model that could be customized by param- eters selection values. A genetic algorithm and other search techniques were employed to rank NCAA Division I-A football with this technique. Deng et al. [11] employed a power law ranking method based on prize monies collected or points earned on 12 different sports. The results for the various sports yielded similar graphs with almost matching exponents for the power law. The PageRank algorithm was developed by Larry Page in 1996 for ranking academic papers. It is a probability distribution that was quickly applied to webpages by use of a weighted network to optimize web searches, where quality of hyperlinks of a webpage produces an advantages PageRank score that pushes the webpage up on search results [12]. Lazova and Basnarkov [13] used FIFA World Cup soccer matches to populate a graph where the PageRank algorithm is applied as an effective ranking system by use of a method to assign weights to the graph based on matches won and goals scored. Tennant et al. [14] produced a general model of the complexity of the sport of boxing exploring match play between welterweight boxers from 2004 to 2015. The PageRank algorithm was used to rank the boxers from a directed graph. The rankings produced were compared with the sport’s notoriously corrupt sanctioning bodies, journalistic rankings and other ranking systems. This work supplied further confirmation of the value of the network-based analysis in athletic match play, but was limited in its timespan and had the narrow scope of only analysing welterweight fighters.

2.1 Pound for pound The approach to deciding who the best-ever in boxing must include a list that consolidates all boxers from all weight classes. Currently, the debate on who is the best of the best within boxing no matter the era is labelled as the P4P champion [15–17]. These lists (i.e. rankings) comprise of fighters who competed directly with other fighters on the lists, and fighters who fought decades prior to other fighters. While there are several different lists publicly available, the majority of those lists come from non-academic sources. Moreover, many of the rubrics used for creating the lists can be argued as biased (e.g. how exciting the WHO WAS THE GREATEST OF ALL-TIME? 3

fighter is) or subjective (e.g. quality of opposition). At any rate, the goal for each ranking is to identify the best P4P fighter; to ever fight. One limitation worth mentioning is the competitive structure professional boxing adopts that influ- ences the different levels of competition a modern-era boxer may face over their career. There are significant factors identified by past scholars [18] that explains reasons why the best boxers seldom fight each other within a particular weight class. The governance structure for professional boxing is not Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 exclusive to one sanctioning body like other individual professional sports (e.g. golf, tennis, bowling). Professional boxing has [at least] four major boxing sanctioning bodies that have significant influence in the industry: World Boxing Council (WBC), World Boxing Association (WBA), International Boxing Federation (IBF) and World Boxing Organization (WBO). Each of these sanctioning bodies claim to have a world champion in each of the respective weight classes [18]. The main implication of having different sanctioning bodies for professional boxing is a lack of ability to identify the best boxer within a weight class. Additionally, there seems to be a lack of oversight in the governance and enforcement of title fights for each sanctioning body, along with corruption within the decisions of fight organizers and promotors [18]. This essentially affects contractional agreements which leads to champion fighters to avoid fighting the best opponents when defending their title. Thus, the limitation exists for the researchers to establish a surface-level boundary to identify which fighters had comparable fighting resumes.

2.2 Past rubrics for all-time best The P4P lists contain substantial popularity for active boxers. The Transnational Boxing Rankings Board [19] has consistently [since 2012] put out the perennial top-10 P4P rankings. They utilize a point-system to attempt to quantify the answers to the following questions to determine their list: how high is the quality of the fighter’s recent opposition? How strong is the fighter’s career-long resume? How advanced is the fighter’s ring generalship? How willing is the fighter to accept all-comers? The authors of this list acknowledge the subjectivity of the list by mentioning the frequent occurrence of a lively debate. When considering the best fighters in history, McRae [15] acknowledged the element of subjectivity in the list for the top 50 P4P boxers of all time. Factors like wins, losses, world titles and the quality of opposition were variables incorporated. The appeal of the list seemed to be the looseness of the list itself, thus allowing for enough ambiguity to spark a discussion. Mulvaney [16] seemed to clarify reasoning for a more objective rubric by including significant variables such as: in-ring performance, achievements, dominance and mainstream appeal. However, even by attempting to quantify these significant variables, the author appears to acknowledge that choosing these particular variables is, in fact, subjective. This list, then, is not just the 50 greatest fighters of all time. It is the 50 fighters who were the greatest in their time [16]. When considering a fighter’s current status, Benson [1] explains that four criteria are considered when ranking fighters P4P. The factors are result, performance, resume and activity. The result of a fight trumps the other factors and essentially is the most influential. The other three factors are equivalent in value. The performance of a fighter is the manner in which a fighter’s result is determined. The resume is simply the opponents the fighter beat, and titles achieved. The activity is the frequency a fighter fights against quality competition. To date, BoxRec [20] appears to quantify the subjective nature of determining the best P4P boxer of all time with their rubrics and methods. Their current ratings structure considers 35 different points of measurement. The ratings structure not only considers performance, results and timing of the fights, but also applies ceilings and limits to the amount a boxer’s rating can differentiate between fights (e.g. 4 A. G. TENNANT ET AL.

The rating of a boxer is reduced by up to 50% in proportion to the difference of two times the rating points of his best opponent in this time period minus his own rating). While BoxRec.com acknowledges that there are inaccuracies and anomalies due to missing historical data, the foundation for their ratings system stands on the unbiased approach to their formulas. However, it appears there still lies ambiguity in the ceilings imposed within the formula, and total points awarded (e.g. 1 point, when a boxer defeats an opponent, who already won a bout against a winning opponent within 18 months) for a number of the Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 criteria points. This gives reason to analyse further using different methods to determine a P4P boxing ranking (e.g. Page Rank).

2.3 Data set BoxRec is an online database that is the preferred source for boxing data post the Marquess of Queensberry Rules. Both male and female, past and present professional boxer’s records are catalogued on this website for the public to explore. There are in excess of over 2.1 million bouts entered into the database that have been researched and validated by global volunteers. Besides the results, the boxing data bout history contained on the site can include venues, referees, judges, official weights, knockout times, promoters and descriptions of many bouts. While small subsets are available categorized by date or boxer, the complete data set is not available for download in its entirety. The researchers approached the staff of the website to obtain the whole data set, and they graciously agreed to send the data set in .csv format. Researchers took a historical perspective emphasizing the significance of the sport’s long history by requesting a data set that included bouts from 1897 to 2019. The investigation that follows utilized two data sets provided by BoxRec.com. These two data sets were already filtered by BoxRec’s scoring system for bouts that involved fighters with at least 10 points or 50 points. This scoring system that BoxRec employs is essentially an algebraic formula that involves several conditions that are checked and then re-calculated daily, incorporating any new bouts into the database. This filtering was needed as a data set involving all 2.1 million bouts would be unmanageable to work with. Nonetheless, the scale of this data involved up to 307,407 bouts and 61,232 boxers and should be considered more than adequate.

3. Methods 3.1 Network creation-directed graph A directed graph which is analogous to hyperlinks (edges) within webpages (nodes) was used by employing the idea into the boxing world by linking fighters through bouts. The researchers started by preprocessing the data in spreadsheet form. This included preparing the data to assist in the creation of the directed graph with some data management clean-up techniques. This involved positioning the data in a source and target arrangement for the nodes to reflect winners and losers making this a directed graph. The edges are the bouts between the fighters that travel from the loser to the winner. The source for the data was the loser of the boxing match and the target was the winner of the bout. A small subset of the entire directed graph can be seen below for understanding in Fig. 1. For illustrative purposes, a subgraph of the entire directed graph was constructed from heavyweight champion Jack Dempsey’s last six fights. This subgraph is made up of six nodes (boxers) and six edges (bouts). The nodes are displayed as the boxer’s images with edges depicted as arrows. If an edge is present between nodes this reflects that a bout was fought between these boxers. For example, Dempsey’s node has an edge entering this node originating from Firpo, Gibbons, Sharkey and Carpentier. These edges represent victories for Dempsey as the target node with the depicted arrow originating from the losing WHO WAS THE GREATEST OF ALL-TIME? 5 Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020

Fig. 1. Directed graph example of Jack Dempsey’s last six ﬁghts.

fighter, or more accurately, defined as the source node. This arrangement gives the graph its directed nature. Conversely, Dempsey shares two other edges with Gene Tunney shown as dotted arrows. These edges are both directed to enter Tunney’s node representing two bouts fought with Dempsey and both being losses. Any boxing matches that were fought to a draw, disqualification or no contest were eliminated from the data set as no one won the match and a directed graph could not use this data. Furthermore, identical edges were combined so that only a single edge could connect a source and a target. An illustrative example of this is seen in Fig. 2 first showing the original arrangement of edges between just two nodes. This example relies on the all-time great saga of fights between Sugar Ray Robinson and Jake LaMotta who fought a staggering six times between 1942 and 1951 with Robinson winning five out of the six. Duplicate edges were combined and then given a weight to reflect that multiple fights status. For example, the original six bouts were combined into two by taking Robinson’s five edges that were wins and converting it to one single edge with a weight of 5. The single LaMotta win was left alone and given a weight of 1. These weights will be utilized when calculating the PageRank values for the individual nodes. The original size of the data sets and the reduced size of the now weighted-directed networks can be seen in Table 1. After the removal of bouts without a winner and the reduction due to weighting the rematches, the networks were both reduced by about 20% of their edges and 1% of their nodes.

3.2 PageRank calculation Once the directed graph was created and the edges are weighted the teams can be ranked. To rank the boxers, the PageRank algorithm was used, which was originally developed in the webpage search engine 6 A. G. TENNANT ET AL. Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020

Fig. 2. Directed graph and modiﬁed weighted-directed graph example of ﬁghts between Sugar Ray Robinson and Jake LaMotta.

Table 1 Network creation statistics

Original Weighted-directed Nodes Edges Nodes Edges 10 pts. 62,033 388,801 61,232 307,407 50 pts. 12,521 58,744 12,425 47,309

revolution of the late 1990’s. The concept is that the webpages are assigned higher PageRank values based not only on the number, but also the quality of the links. Page et al. (1999) gave the PageRank algorithm in early publications as shown in equation 1: In terms of boxers, PR(A) is the PageRank of boxer A, PR(Ti) is the PageRank of boxer Ti linked to boxer A with a loss, C(Ti) is the number of edges leaving boxer Ti; that is, number of losses. Finally, d is a damping factor, empirically preferred to be 0.85 [14]. The algorithm is an iterative process that can be explained by the concept of a random walk through the directed graph. The walker will randomly enter the graph on any source node (boxer) then randomly travel along one of the outward-bound edges (match that was lost) to the target node (winning boxer). This process has been modiﬁed in our directed graph to favour certain edges based on the weights discussed previously. This gives the ability to place more emphasis on those edges that represent multiple wins. For instance, if a weight of 3 was present on a particular edge leaving a node, that would give that edge a three times more likely chance of being selected opposed to an edge leaving that node weighted with WHO WAS THE GREATEST OF ALL-TIME? 7

Table 2 PageRank top1 10 P4P boxers from 1897 to 2019 for 50 point data set

Rank First name Last name Career start Career end PageRank 1 Sugar Ray Robinson 4 October 1940 11 October 1965 0.00242 2 Evander Holyﬁeld 15 November 1984 7 May 2011 0.00211 3 Floyd Mayweather Jr. 11 October 1996 26 August 2017 0.00206 Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 4 Emile Grifﬁth 2 June 1958 30 July 1977 0.00204 5 Manny Pacquiao 22 January 1995 NA 0.00202 6 Harry Greb 29 May 1913 19 August 1926 0.00190 7 Muhammad Ali 29 October 1960 11 December 1981 0.00189 8 Archie Moore 3 September 1935 15 March 1963 0.00187 9 Ezzard Charles 12 March 1940 1 September 1959 0.00174 10 Lennox Lewis 27 June 1989 21 June 2003 0.00163

a 1. This random walk will continuously occur and essentially build a probability distribution on what node (boxer) the walker would be located if the algorithm was stopped. The algorithm was run on both the smaller quantity of bouts in the 50-point and larger 10-point data sets. To force convergence of the PageRank algorithm, a damping factor was utilized, this is often referred to as teleportation when applied to the random walk paradigm. For instance, with a damping factor of d = 0.85 on the current node position the walker will 15% of the time not take an outward-bound edge and randomly teleport to another node (boxer) in the directed graph. This process of teleportation avoids the walker being trapped on a node with no edges out (i.e. undefeated boxer), commonly referred to as a dangling node in network science, in this technique non-convergence of the algorithm is avoided. In this study, the standard damping factor of 0.85 was modiﬁed for larger 10-point data set to both 0.90 and 0.95 to explore the variance in the results. Alternatively, the Eigenvector centrality was also calculated for the network to compare later to the results to help explain the advantages of using PageRank. Centrality has been historically used in various networks as an evaluation of individual node importance. The method employed in this research work departs from Tennant et al. [14] in the much larger historical data set, the inclusion of all weight classes, and the use of weighted edges.

4. Results and discussion Utilizing the PageRank algorithm can provide an unbiased way to rank boxers based solely on their PageRank score. For both the smaller 50-point and larger 10-point data set networks the top 100 boxers according to their PageRank are presented in the subgraph visualization displayed in Figs 3 and 4. These directed graphs are filtered by only including the top 100 PageRank results and edges associated between these nodes. The larger diameter and darker green colour nodes represent the higher PageRank values. The top 10 boxers by PageRank score have been labeled in both visualizations to display the results in a relatable form. Additionally, Tables 2 and 3 gives a more detailed view of these results with some supplementary career data for 50- and 10-point data set networks, respectively. This depiction in Figs 3 and 4 assisted in understanding the significance of the analysis. For instance, Archie Moore in both graphs has many edges leaving or entering his node. Which reflects the historical fact that he was not only an extremely active fighter with 220 bouts, but one that was fighting a high degree 8 A. G. TENNANT ET AL.

Table 3 PageRank top 10 P4P boxers from 1897 to 2019 for 10 point data set

Rank First name Last name Career start Career end PageRank 1 Floyd Mayweather Jr. 11 October 1996 26 August 2017 0.00143 2 Manny Pacquiao 22 January 1995 NA 0.00129 3 Sugar Ray Robinson 4 October 1940 10 November 1965 0.00104 Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 4 Archie Moore 3 September 1935 15 March 1963 0.00098 5 Emile Grifﬁth 2 June 1958 30 July 1977 0.00089 6 Muhammad Ali 29 October 1960 11 December 1981 0.00086 7 Harry Greb 29 May 1913 19 August 1926 0.00084 8 Wladimir Klitschko 16 November 1996 29 April 2017 0.00083 9 Evander Holyﬁeld 15 November 1984 7 May 2011 0.00076 10 Bernard Hopkins 11 October 1988 17 December 2016 0.00075

of opposition since these edges appear in this subgraph with only high PageRank boxers. Additionally, these graphs tend to show the clustering of fighters by their weight class and era they fought. For instance, Harry Greb in both subgraph figures, is set aside from the other top 10 fighters. This is due to the era he fought in (i.e. the early 1900s) being significantly different than the eras of the other nine fighters. By inspecting the list of names provided by the PageRank analysis in Table 2, it becomes immediately evident to even the casual fan of the sport, the power of the technique to rank boxers. The highest ranked boxer, Sugar Ray Robinson, is a well-known pugilistic wizard of the golden era of boxing and is often considered by journalist and fans as the greatest middleweight boxer of all time. Robinson wore the crown of the world middleweight title on five occasions. His career was prolific with a record of 174 victories, 19 defeats and 6 draws from debuting in 1940 and entering the ring for the final time in 1965. Robinson was a well-known celebrity in his time and had gained the respect of the American public for his speed, power, ring generalship, toughness and graceful style [21]. Further down the rankings, the only active fighter on this list, Manny Pacquiao, appears at the number five position. Given the moniker of the Pac Manž due to his record of gobbling up opponents his current record stands at 62 victories, 7 defeats and 2 draws and has been active since 1995. Pacquiao had success early on as a champion as a flyweight (112 lb), super bantamweight (122 lb), super featherweight (130 lb), lightweight (135 lb) and currently as a welterweight (147 lb). This success across multiple weight classes connected Pacquiao with many other high-ranking nodes within the directed graph creating a scenario where higher PageRank scores can be expected [22]. As a comparison to the PageRank results, Table 4 displays the Eigenvector centrality results. The names of the top 10 boxers from this analysis are fairly recognizable to any devotee of the history of the sport. Henry Armstrong held the top spot followed by Sugar Ray Robinson, two of the most decorated boxers in the history of the sport, both multi-divisional champs. Though upon a closer look of these results it was discovered that these top 10 spots were completely made up of boxers who haven’t fought in at least 50 years. More specifically, a timeline clustering of the Eigenvector centrality results was noticeable. The list was predominantly comprised of boxers who fought in the 1930s and 40s. This is interesting as this is considered by historians to be the era known as the golden age of boxing. Nevertheless, it is highly unlikely that any list of the best of all time should be that biased towards a specific era. Additionally, the list of the top 10 was exclusively composed of lightweights, welterweights and middleweights, the WHO WAS THE GREATEST OF ALL-TIME? 9 Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020

Fig. 3. Weighted directed subgraph ﬁltered by top 100 boxers by PageRank value from 50-point data set.

high value Eigenvector centrality results completely excluded ﬂyweights, bantamweights, featherweights, light heavyweights, cruiserweights and heavyweights. It is improbable that the exclusions of all these weight classes accurately represent the history of the sport in any all-time P4P ranking list. The difference between the PageRank and Eigenvector centrality is the inclusion of the teleportation phenomenon in the PageRank algorithm. This teleportation phenomena explains the superior wide-ranging results of the PageRank algorithm. Where the Eigenvector centrality identiﬁed the broader era, when boxing was potentially at its peak. The PageRank results capture the individual boxer’s career achievements superiorly. For a percentage of the iterations of the analysis, the act of teleporting randomly to another node (boxer) likely outside of any cluster the node belongs to prevents the results from being 10 A. G. TENNANT ET AL. Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020

Fig. 4. Weighted directed subgraph ﬁltered by top 100 boxers by PageRank value from 10-point data set.

Table 4 Eigenvector centrality top 10 P4P boxers from 1897 to 2019 for 50 point data set

Rank First name Last name Career start Career end E. centrality 1 Henry Armstrong 27 July 1931 14 February 1945 1.00000 2 Sugar Ray Robinson 4 October 1940 10 November 1965 0.96297 3 Tony Canzoneri 2 July 1925 1 November 1939 0.76707 4 Wesley Ramey 14 August 1929 20 June 1941 0.75452 5 Cocoa Kid 27 May 1929 24 August 1948 0.71421 6 Holman Williams 12 August 1932 30 June 1948 0.71405 7 Fritzie Zivic 5 October 1931 17 January 1949 0.69543 8 Sammy Angott 9 March 1935 8 August 1950 0.69104 9 Ike Williams 15 March 1940 12 August 1955 0.62509 10 Lou Ambers 9 June 1932 28 February 1941 0.60380

centralized to any specific era or grouping of weight classes in the sport. This spreads the higher ranks out to various weight classes and to a considerably less biased time frame. In Table 5, the results of modifying the damping factor for the larger 10-point network is presented. Floyd Mayweather Junior does stay in the top spot for each of these rankings, but one noticeable trend took place. Successful heavyweight boxers moved up in the rankings as the damping factor is increased, the most noticeable being Klitschko, who in the 0.85 analysis was ranked 8 but at the 0.95 analysis had climbed to the number 2 position. Within these three separate rankings for the top 10 fighters, the heavyweight fighters either climbed in rankings or remained constant. WHO WAS THE GREATEST OF ALL-TIME? 11

Table 5 The ranking results with varying PageRank damping factors

PageRank algorithm damping factors Rank 0.85 0.9 0.95 Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 1 Mayweather Jr. Mayweather Jr. Mayweather Jr. 2 Pacquiao Pacquiao Klitschko 3 Robinson Robinson Ali 4 Moore Moore Moore 5 Griffith Griffith Robinson 6 Ali Ali Pacquiao 7 Greb Klitschko Holyfield 8 Klitschko Holyfield Griffith 9 Holyfield Greb Charles 10 Hopkins Charles Lewis

The assumption behind this phenomenon is that boxers tend to move up in weight due to age or looking for a big money fight. When a naturally smaller boxer moves up in weight to fight a larger boxer he is at a disadvantage and will be more likely to lose. This basic fact will eventually produce more edges leading to larger boxers eventually terminating at a heavyweight boxer. The lower the damping factor measures, the more likely the random walker is to be teleported and not be directed along this trend. Since this is a P4P list and the researchers are attempting to avoid bias based on the size of prize fighter, the 0.85 damping factor is preferred as it lessens this tendency. In Table 6, the PageRank results for the 10-point data set with a damping factor of 0.85 are compared with the corresponding rank from BoxRec’s all-time list. The top two spots directly match with May- weather and Pacquiao, validating the high skill level of both these fighters who have been active the last 20 years. A two-dimensional plot containing the BoxRec rankings and the PageRank results is shown in Fig. 5 a rather weak correlation is shown between the two through a simple linear curve fit. Nevertheless, this visualization is for data exploration which helps with the corresponding table to identify the presence of outliers to investigate. Taking an inquisitive look at those rankings with the highest discrepancy boxers like Holman Williams and Cocoa Kid seem to be ranked either excessively high or low by the PageRank algorithm or BoxRec, respectively. A thorough inspection of these fighter’s data shows that Williams and Kid fought a staggering 12 times against each other, an anomaly even before the 1960’s when rematches were more common. The high weight on these edges makes them much more likely to be selected during the random walk thus potentially skewing the results. A limit could be worked into the network to reduce excessively high weights due to rematches to a prescribed max weight. It is worth noting that Williams and Kid are highly reputable all-time great fighters that have been inducted into the International Boxing Hall of Fame [23]. Maxie Rosenbloom and Henry Armstrong are another two outliers in the linear curve fit. First exploring Rosenbloom’s career, it is remarkable to see he fought a staggering 274 times from 1923 to 1939 which overlaps with the timeframe of the sport’s golden age. Nicknamed Slapsie Maxie due to his softer open fist punching style he amassed a record of 207 wins, 39 loses and 26 draws. Rosenbloom is considered one of the most elusive fighters to punch in the history of the sport [21]. His high degree of activity 12 A. G. TENNANT ET AL.

Table 6 Comparison of PageRank and associated BoxRec ranking

Rankings Boxer PageRank BoxRec Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 Floyd Mayweather Jr. 1 1 Manny Pacquiao 2 2 Sugar Ray Robinson 3 5 Archie Moore 4 8 Emile Grifﬁth 5 43 Muhammad Ali 6 4 Harry Greb 7 19 Wladimir Klitschko 8 36 Evander Holyﬁeld 9 11 Bernard Hopkins 10 6 Ezzard Charles 11 63 Roy Jones Jr. 12 20 Saul Alvarez 13 50 Andre Ward 14 33 Maxie Rosenbloom 15 342 Holman Williams 16 408 Dick Tiger 17 66 Henry Armstrong 18 96 Cocoa Kid 19 923 Lennox Lewis 20 79

Fig. 5. BoxRec rankings vs. PageRank positions. and success in this timeframe with higher eigenvector centrality values correspondingly explain his high PageRank score. WHO WAS THE GREATEST OF ALL-TIME? 13

Table 7 Comparison of PageRank and various boxing journalist rankings

Rank PageRank Kieran Mulvaney (2007) Bert Sugar (1988) Kevin McRae (2012) 1 Floyd Mayweather Jr. Sugar Ray Robinson Sugar Ray Robinson Sugar Ray Robinson 2 Manny Pacquiao Muhammad Ali Henry Armstrong Henry Armstrong 3 Sugar Ray Robinson Henry Armstrong Harry Greb Willie Pep Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 4 Archie Moore Joe Louis Jack Dempsey Muhammad Ali 5 Emile Grifﬁth Willie Pep Benny Leonard Joe Louis 6 Muhammad Ali Roberto Duran Joe Louis Roberto Duran 7 Harry Greb Benny Leonard Mickey Walker Jack Johnson 8 Wladimir Klitschko Jack Johnson Sam Langford Jack Dempsey 9 Evander Holyﬁeld Jack Dempsey Tony Canzoneri Benny Leonard 10 Bernard Hopkins Sam Langford Muhammad Ali Sugar Ray Leonard 11 Ezzard Charles Joe Gans Joe Gans Harry Greb 12 Roy Jones Jr. Sugar Ray Leonard Willie Pep Joe Gans 13 Saul Alvarez Harry Greb Jack Johnson Sam Langford 14 Andre Ward Rocky Marciano Barney Ross Gene Tunney 15 Maxie Rosenbloom Jimmy Wilde Jimmy Wilde Rocky Marciano 16 Holman Williams Gene Tunney Gene Tunney Archie Moore 17 Dick Tiger Mickey Walker Roberto Duran Jimmy Wilde 18 Henry Armstrong Archie Moore Johnny Dundee Mickey Walker 19 Cocoa Kid Stanley Ketchel Rocky Marciano Julio Cesar Chavez 20 Lennox Lewis George Foreman Joe Walcott George Foreman

Reviewing the career of Henry Armstrong a fighter who was ranked 96th by BoxRec and 18th by the authors gives further proof of the validity of the PageRank Algorithm. As will be seen below Armstrong is considered one of the all-time greats by historians of the sport, he is the sole boxer to simultaneously hold world championships in three weight classes (feather-, light- and welter-weight). Nicknamed Homicide Hank he was the polar opposite to Rosenbloom crafty style by fighting with a come forward reckless hard hitting style [21]. He also was active in the timeframe 1931–1945 that overlapped with the sport’s golden age that encapsulated the higher eigenvector centrality values for that era. BoxRec 96th rank is rather low for such an accomplished boxer, the PageRank results align more accurately with this higher rank expectation based on the historical significance of Armstrong’s career. Table 7 takes a comparative look at the 10-point data set with a damping factor of 0.85 PageRank listings with three separate journalists top 20 boxers. These three lists have been composed by highly respected boxing writers and historians, with Burt Sugar’s list the most highly esteemed [15–17]. These lists have many similarities and it should be remembered that they were not created in a vacuum and were likely impacted by each other either consciously or subconsciously. All three lists are highly skewed with 15 out of the 20 boxers fighting before the 1960’s. Some boxers on the list have very little film of their events and their rankings are based on second-hand accounts. It is highly likely that these lists are falling under an overly nostalgic look on a period when the sport of boxing was far more culturally significant. These boxers were household names and commanded the world’s attention with demigod like status. A qualitative non-analytic list would be hard pressed to ignore this and not over emphasize a boxer’s career or era. 14 A. G. TENNANT ET AL.

The PageRank listings are slightly skewed to boxers who fought after the 1980’s with 9 out of the 20 boxers coming from this time frame. This result is likely caused by the fact that as boxer’s age they are more likely to lose as they run into younger opposition. This would have a cascading effect and tend to push higher PageRank scores in a directed graph to more current boxers. Further research could use modiﬁed weights adjusted to place more emphasis on wins from the past by use of factors from a timeline scale. In general, the PageRank results had signiﬁcantly better distribution of boxers from the many eras Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 included in the analysis than previous journalistic rankings.

5. Equations

(PR(T ) (PR(T ) PR(A) = (1 − d) + d 1 + ... + n (1) C(T1) C(Tn)

Acknowledgements The authors acknowledges the support of the BoxRec.com and especially Martin Reichert within that organization.

References

1. Benson, M. (2019) Ranked top ten pound-for-pound boxers in the world, including vasyl lomachenko and canelo alvarez. https://talksport.com/sport/boxing/466713/top-ten-pound-for-pound-boxers- vasyl-lomachenko-canelo/, August 2019. 2. Batty, M. (2007) Cities and Complexity: Understanding Cities with Cellular Automata, Agent-based Models, and Fractals. Cambridge, MA: The MIT Press. 3. Batty, M. (2013) The New Science of Cities. Cambridge, MA: MIT Press. 4. Derrible, S. (2017) Urban infrastructure is not a tree: integrating and decentralizing urban infrastructure systems. Environ. Plan. B, 44, 553–569. 5. Peiravian, F. & Derrible, S. (2017) Multi-dimensional geometric complexity in urban transportation systems. J. Transp. Land Use, 10. 6. Derrible, S. & Kennedy, C. (2010) The complexity and robustness of metro networks. Physica A, 389, 3678–3691. 7. Sturmberg, J. P. & Martin, C. (2013) Handbook of Systems and Complexity in Health. New York, NY: Springer Science & Business Media. 8. Rual, J.-F., Venkatesan, K., Hao, T., Hirozane-Kishikawa, T., Dricot, A., Li, N., Berriz, G. F., Gibbons, F. D., Dreze, M. & Ayivi-Guedehoussou, N. (2005) Towards a proteome-scale map of the human protein protein interaction network. Nature, 437, 1173. 9. Borge-Holthoefer, J., Banos, R. A., Gonz´alez-Bail´on, S. & Moreno, Y. (2013) Cascading behaviour in complex socio-technical networks. J. Complex Netw., 1, 3–24. 10. Cassady, C. R., Maillart, L. M. & Salman, S. (2005) Ranking sports teams: a customizable quadratic assignment approach. INFORMS J. Appl. Anal., 35, 497–510. 11. Deng, W., Li, W., Cai, X., Bulou, A. & Wang, Q. A. (2012) Universal scaling in sports ranking. N. J. Phys., 14, 093038. 12. Page, L., Brin, S., Motwani, R. & Winograd, T. (1999) The PageRank citation ranking: bringing order to the web. Technical Report. 13. Lazova, V. & Basnarkov, L. (2015) PageRank approach to ranking national football teams. arXiv preprint arXiv:1503.01331. WHO WAS THE GREATEST OF ALL-TIME? 15

14. Tennant, A. G., Ahmad, N. & Derrible, S. (2017) Complexity analysis in the sport of boxing. J. Complex Netw., 5, 953–963. 15. McRae, K. (2012) Bleacher report: the top 50 pound-for-pound boxers of all time. https://bleacherreport.com/ articles/1436191 December 2012. 16. Mulvaney, K. (2007) Espn:50 greatest. http://www.espn.com/sports/boxing/greatest/featureVideo?

page=greatest4150, May 2007. Downloaded from https://academic.oup.com/comnet/article-abstract/8/1/cnaa009/5770924 by guest on 02 March 2020 17. Sugar, B. R. (1988) 100 Greatest Boxers of All Time, revised edition. New York, NY: Crescent. 18. Andref, W. & Szymanski, S. (2006) Handbook on the Economics of Sport. Cheltenham, UK: Edward Elgar Publishing. 19. Transnational Boxing Rankings Board. (2019) P4P. http://www.tbrb.org/p4p/, September 2019. 20. BoxRec. (2019) Boxrec ratings description. https://boxrec.com/media/index.php/BoxRec_Ratings_Description, September 2019. 21. Roberts, J. B., Skutt, A. G. & International Boxing Hall of Fame (2006) The Boxing Register: International Boxing Hall of Fame Ofﬁcial Record Book. Boxing Register. Ithaca, NY: McBooks Press. 22. BoxRec. (2019) Manny pacquiao’s boxing record. https://boxrec.com/en/proboxer/6129, July 2019. 23. International Boxing Hall of Fame. (2019) Inductees. http://www.ibhof.com/pages/about/inductees/ inducteeindex.html, June 2019.